#### Abstract

We present the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVIFOWCS) measure in this paper, which combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator. The main advantage of the IVIFOWCS measure provides a parameterized family of similarity measures, and the decision maker can use the IVIFOWCS measure to consider a lot of possibilities and select the aggregation operator in accordance with his interests. We have studied some of its main properties and particular cases such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity (IVIFOWACS) measure and the interval-valued intuitionistic fuzzy maximum cosine similarity (IVIFMAXCS) measure. The IVIFOWCS measure not only is a generalization of some similarity measure, but also it can deal with the correlation of different decision matrices for interval-valued intuitionistic fuzzy values. Furthermore, we present an application of IVIFOWCS measure to the group decision-making problem. Finally the existing similarity measures are compared with the IVIFOWCS measure by an illustrative example.

#### 1. Introduction

The similarity measure is an important tool for measuring the degree of similarity between two objects, which is very useful in some areas, such as decision-making, machine learning, pattern recognition, and medical diagnosis [1–6]. Over the past several decades, a variety of similarity measures have been introduced and investigated [7–14] based on intuitionistic fuzzy sets (IFSs) [15]. For example, Li and Cheng [9] investigated similarity measures on IFSs and showed how these measures may be used in pattern recognition problems. Later, Liang and Shi [10] introduced several new similarity measures on IFSs and discussed the relationships between these measures. Hung and Yang [11] presented a method to calculate the distance between IFSs based on the Hausdorff distance and used this distance to generate several similarity measures between IFSs. Furthermore, Hung and Yang [12] presented two new similarity measures between IFSs, which have been found to satisfy some similarity measure axioms. One of many similarity measures is the cosine similarity measure based on Bhattacharyya’s distance [13], which is defined as the inner product of two vectors divided by the product of their lengths. Ye [14] proposed a cosine similarity measure between IFSs and applied it to medical diagnosis and pattern recognition. However, in some cases, the degrees of membership or nonmembership are sometimes assumed not exactly as a number but as a whole interval; Atanassov and Gargov [16] introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs). Furthermore, Xu [17] developed some similarity measures of intuitionistic fuzzy sets and applied them to pattern recognition. Ye [18] proposed a cosine similarity measure for IVIFSs and applied it to multiple attribute decision-making problems.

When similarity measures are widely used in decision-making problems, the importance of ordered position of each degree of similarity should be emphasized. In other words, the higher the degree of similarity, the higher the weight which should be assigned to it; a very useful technique is the ordered weighted averaging (OWA) operator. The OWA operator is introduced by Yager [19], which is a very well-known aggregation operator that provides a parameterized family of aggregation operators including the maximum, the minimum, and the average as special cases. The prominent characteristic of the OWA operator is the reordering step. Since it has appeared, the OWA operator has been widely extended to other aggregation environments, including linguistic environment (Merigó and Casanovas [20], Wei and Zhao [21], and Zhou and Chen [22, 23]), fuzzy environment (Merigó and Gil-Lafuente [24], Xu [25]), intuitionistic fuzzy environment (Li [26], Zeng and Su [27], and Zhou et al. [28, 29]), and interval-valued intuitionistic fuzzy environment (Li et al. [30], Yu et al. [31], and Zhou et al. [32]), and used in areas such as decision-making and neural networks (Yager [33], Merigó and Gil-Lafuente [34], and Zhou et al. [35–38]).

The aim of this paper is to introduce the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVIFOWCS) measure. It combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized OWA operator. A more complete formulation of the cosine similarity measure is obtained because it can consider parameterized families of operators that include the maximum, the minimum, and the average as special cases. Using the advantage of IVIFOWCS measure can relieve the influence of unduly large or unduly small deviations on the aggregation results. This measure provides a robust formulation that includes a wide range of particular cases, such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity (IVIFOWACS) measure, the interval-valued intuitionistic fuzzy ordered weighted quadratic cosine similarity (IVIFOWQCS) measure, the interval-valued intuitionistic fuzzy ordered weighted geometric cosine similarity (IVIFOWGCS) measure, the interval-valued intuitionistic fuzzy maximum cosine similarity (IVIFMAXCS) measure, the interval-valued intuitionistic fuzzy minimum cosine similarity (IVIFMINCS) measure, the interval-valued intuitionistic fuzzy normalized cosine similarity (IVIFNCS) measure, the interval-valued intuitionistic fuzzy normalized arithmetic cosine similarity (IVIFNACS) measure, and the interval-valued intuitionistic fuzzy normalized geometric cosine similarity (IVIFNGCS) measure. The decision maker is able to consider a wide range of scenarios and select the one that is in accordance with his interests.

The paper is organized as follows. In Section 2, we briefly review the concepts of IFSs, IVIFSs, the cosine similarity measure for IVIFSs, and the OWA operator. In Section 3, we introduce the IVIFOWCS measure; some properties and different families of the IVIFOWCS measures are analyzed. Section 4 develops an application in the group decision-making problem. Section 5 gives a numerical example. Section 6 summarizes the main conclusions of the paper.

#### 2. Preliminaries

##### 2.1. Basic Concepts of IFSs and IVIFSs

*Definition 1. *Let be a finite universal set; IFs in is defined aswhere are the membership function and nonmembership function, respectively, such that .

Assume ; then is called the hesitation degree of whether belongs to or not. It is obvious that . For convenience, we call an intuitionistic fuzzy number (IFN) and denote the module of as .

*Definition 2. *Let ; IVIFs in is defined as , where intervals and denote the membership degree and nonmembership degree of the element to the set , respectively. For each , the hesitancy degree of an interval intuitionistic fuzzy set is defined as follows: An interval-valued intuitionistic fuzzy number (IVIFN) ; we denote the module of as .

Let and be two IVIFNs; the operations are defined as follows (Ye [18]):(1)(2)(3) if and .

##### 2.2. The OWA Operator

The OWA operator is an aggregation operator that provides a parameterized family of aggregation operators that includes the maximum, the minimum, and the average as special cases. It can be defined as follows.

*Definition 3. *An OWA operator of dimension is a mapping OWA: that has an associated weighting vector with and , such that , where is the largest th of the arguments

Note that the OWA operator is commutative, monotonic, bounded, and idempotent.

Yager [39] developed the generalized OWA (GOWA) operator, which is defined as follows.

*Definition 4. *A GOWA operator is a mapping GOWA: that has an associated weighting with and , and a parameter and , such that where is the largest th of the arguments .

We know that the GOWA operator is also commutative, monotonic, bounded, and idempotent (Yager [39]). We can obtain a group of particular cases. For example, if , then the GOWA operator is reduced to the OWA operator. If , the ordered weighted geometric averaging (OWGA) operator is obtained. If , the ordered weighted harmonic averaging (OWHA) operator is formed.

##### 2.3. Cosine Similarity Measures for IVIFSs

*Definition 5. *Let , assume that there are two IVIFSs and , and a cosine similarity measure between two IVIFSs and is defined as follows:

The cosine similarity measure between and satisfies the following properties:(1)(2)(3) if , i.e., , , and . .

#### 3. Interval-Valued Intuitionistic Fuzzy Ordered Weighted Cosine Similarity Measure

In this section, we will introduce the IVIFOWCS measure, which is a similarity measure that uses the cosine similarity measure for IVIFS in the GOWA operator.

##### 3.1. The IVIFOWCS Measure

Let , be two interval-valued intuitionistic fuzzy matrices, , are IVIFNs for all , and assume that and for . We can define the IVIFOWCS measure as follows.

*Definition 6. *An IVIFOWCS measure of dimension is a mapping IVIFOWCS: that has an associated weighting vector with and , such that where is the cosine similarity measure between IVIFS and and is any permutation of , such that

*Remark 7. *If in and , the IVIFOWCS measure reduces to the cosine similarity measure for IVIFS (Ye [18]).

*Example 8. *LetBy (4), we can get , , and .

Then , , and .

If , by using the IVIFOWCS measure, we can obtain the cosine similarity measures corresponding to some special cases of the parameter , which are shown in Table 1.

##### 3.2. Properties of the IVIFOWCS Measure

The IVIFOWCS measure is commutative, monotonic, bounded, idempotent, nonnegative, and reflexive. These properties are shown with the following theorems.

Theorem 9 (commutativity-GOWA aggregation). *Let and .**If is any permutation of the arguments , then .*

*Proof. *Let Because is a permutation of the arguments , and we know for all , we have . Then we complete the proof of Theorem 9.

Theorem 10 (commutativity-similarity measure). *Let ; then*

*Proof. *Let Because , then for all .

This completes the proof of Theorem 10.

Theorem 11 (monotonicity). *Let , , and ; if for all , then*

*Proof. *LetBecause , then for all .

This completes the proof of Theorem 11.

Theorem 12 (boundary). *Let and for all ; then*

*Proof. *If and , noticing , thenThen we complete the proof of Theorem 12.

Theorem 13 (idempotency). *Let and ; if ( is a constant) for all , then .*

*Proof. *LetBecause for all , we have , .

Then .

Theorem 14 (nonnegativity). *Let and ; then*

*Proof. *It is straightforward and thus omitted.

Theorem 15 (reflexivity). *Let ; then .*

*Proof. *Let Because for all , then

This completes the proof of Theorem 15.

##### 3.3. Families of the IVIFOWCS Measures

By using different cases of the weighting vector and parameter , we are able to obtain a wide range of particular types of the IVIFOWCS measure.

###### 3.3.1. Analyzing the Parameter

By choosing different cases of the parameter in the IVIFOWCS measure, we can obtain different types of cosine similarity measure, such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity (IVIFOWACS) measure, the interval-valued intuitionistic fuzzy ordered weighted quadratic cosine similarity (IVIFOWQCS) measure, and the interval-valued intuitionistic fuzzy ordered weighted geometric cosine similarity (IVIFOWGCS) measure.

*Remark 16. *If , then the IVIFOWCS measure is reduced to IVIFOWACS measure:where is any permutation of , such that

*Remark 17. *If , then the IVIFOWCS measure becomes the IVIFOWQCS measure:where is any permutation of , such that ,

*Remark 18. *If , then the IVIFOWCS measure is reduced to the IVIFOWGCS measure:where is any permutation of , such that ,

###### 3.3.2. Analyzing the Weighting Vector

By choosing a different manifestation of the weighting vector in the IVIFOWCS measure, we are able to obtain different types of cosine similarity measures, such as IVIFMAXCS measure, IVIFMINCS measure, IVIFNCS measure, IVIFNACS measure, and IVIFNGCS measure.

*Remark 19. *If and for all , the IVIFOWCS measure is reduced to IVIFMAXCS measure.

*Remark 20. *If and for all , the IVIFOWCS measure is reduced to IVIFMINCS measure.

*Remark 21. *If for all , the IVIFOWCS measure is reduced to IVIFNCS measure.

Specially, if , we can get the IVIFNACS measure; if , we can get the IVIFNQCS measure; if , the IVIFNCS measure is reduced to IVIFNGCS measure.

#### 4. Multiple Attribute Group Decision-Making with the IVIFOWCS Measure

In this paper, we consider a decision-making application of the IVIFOWCS measure in the selection of investments under uncertainty. Let be a set of alternatives and be the set of attributes. Let be the set of decision makers. Each decision maker provides his own payoff matrix , where is given by the decision maker , for the alternative , with respect to the attribute .

Then based on the IVIFOWCS measure, we propose a method with the IVIFOWCS measure in group decision-making, which involves the following steps.

*Step 1. *Form the ideal alternative by giving the ideal levels of each characteristic, which is shown in Table 2, where is the ideal characteristic of

*Step 2. *Calculate the IVIFCS measure between each preference vector provided by the decision maker and ; the formula is given as follows:where =.

*Step 3. *Utilize the IVIFOWCS measure to aggregate the IVIFCS measure into the collective value of the alternative , where is any permutation of , such that

*Step 4. *Rank all the alternatives in accordance with the collective values in descending order and select the best one of them.

#### 5. Illustrative Example

##### 5.1. An Illustration of the Proposed IVIFOWCS Measure

In the following, we are going to develop a brief example of the new approach in a group decision-making problem about investment selection.

Assume a decision maker wants to invest money in a company; after analyzing the market, he considers six possible alternatives:(1)Invest in a chemical company called (2)Invest in a food company called (3)Invest in a computer company called (4)Invest in a car company called (5)Invest in a furniture company called (6)Invest in a pharmaceutical company called

The decision maker has brought together a group of experts. Each expert reviews the information of the investments from six characteristics : : benefits in the short term : benefits in the mid term : benefits in the long term : risk of the investment : difficulty of the investment : other factors

Three experts provide their opinions about the investments; the results are shown in Tables 3–5.

According to their objectives, the company experts establish the ideal investments, shown in Table 6.

Utilize (4) to calculate the cosine similarity measure between each decision maker’s preference vector and his ideal preference vector ; then we obtain the in Table 7.

Using (23) to aggregate cosine similarity measure into the collective value of all the alternative . For convenience, we assume that the experts’ weighting vector and ; we develop different methods based on the IVIFOWCS measure for selection of an investment. For example, we consider IVIFMAXCS, IVIFMINCS, IVIFNACS, IVIFNQCS, and IVIFNGCS measures. The results are shown in Table 8.

As we can see, the best alternative is . That is to say, the optimal alternative for the investor is the chemical company. Depending on the particular cases of the IVIFOWCS measure used, the ordering of the companies is different. Therefore, which investment will be selected for the decision maker may also differ. We establish an ordering of the investments for some particular case in Table 9.

##### 5.2. A Comparison Analysis with the Existing Method Using IFOWCS Measure

Furthermore, in order to demonstrate the reasonability of the IVIFOWCS measure, we use the intuitionistic fuzzy ordered weighted cosine similarity (IFOWCS) measure to solve the same illustrative example. For comparison, the transformation from IVIFNs to IFNs is carried out by substituting each interval value with the mean value of its upper and lower limits. The values of the illustrated example that have been converted are shown in Tables 10–13.

Zhou et al. [28] introduced the IFOWCS measure. Now, we use the IFOWCS measure to aggregate the collective value of all the alternative . For comparison, we still assume that the experts’ weighting vector and , and the different particular cases of the IFOWCS measure for the aggregation results are shown in Table 14.

We establish an ordering of the alternatives for each special IFOWCS measure; the results are shown in Table 15.

Obviously, the first alternative in each ordering is still the optimal choice; this reveals the validity of the proposed method in this paper. And we see that each aggregation measure may also lead to different results in the IFOWCS measure.

In the end, we analyze how the different parameter value plays a role in the IVIFOWCS measure. Considering different value of : , the collective overall values of six alternatives are shown in Figure 1.

#### 6. Conclusions

In this paper, we presented the IVIFOWCS measure by combining the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized OWA operator, which is very useful to deal with the decision information under uncertain situations. Moreover, we have studied some of its main properties and presented a numerical example of the new approach to see the application of the IVIFOWCS measure in an investment decision-making problem. The main advantage of the IVIFOWCS measure provides a parameterized family of aggregation operators and similarity measure. In addition, the IVIFNs used in this paper are suitable for expressing evaluation; the decision maker can use the IVIFOWCS measure to consider a lot of possibilities and select the aggregation operator that is in accordance with his interests. In future research, we expect to develop further extensions by adding new characteristics in the problem such as probabilistic aggregations.

#### Competing Interests

The authors declare no conflict of interests regarding the publication for the paper.

#### Acknowledgments

This research is fully supported by the Key International Collaboration Project of the National Nature Science Foundation of China (no. 71210003), a grant from National Natural Science foundation of Hunan (2015JJ6041), National Natural Science Foundation of China (11501191), and National Social Science Fund of China (15BTJ028).